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A Semester of First Year Physics with Peter Eyland

Lecture 13 (Pure R,C and L in AC)

In this lecture the following are introduced:
• The average or D.C value
• A pure resistor in an A.C. circuit
• The power dissipated in a resistor
• The effective or r.m.s. value
• A pure capacitor in an A.C. circuit
• The power in a capacitor
• A pure inductor in an A.C. circuit
• The power in an inductor

The Average or D.C value

The average or D.C. value of a periodically varying function is given by the sum over one period divided by the time for one period, as written mathematically:

equation


Example

For the half-wave rectified sine potential shown. Find the average value of the potential

equation


The angular frequency, ω, is related to the period, T, by ω = 2π/T, where π is the Greek letter pi.

equation


Example

For the full-wave rectified sine potential shown. Find the average value of the potential.

equation


equation


The half and full-wave rectified e.m.f.s shown above, are D.C., because the average value is not zero and the current always flows in the same direction. An A.C. has to change direction, as shown in the diagram. E.m.f.s may be represented as the superposition of a steady D.C. component (the average value) and an A.C. component which changes direction.

equation


A Resistor in an AC circuit

equation

The complex applied emf is Emexp[jωt].

The complex potential difference across the resistor is equal to the complex applied emf.

equation


The complex impedance of a circuit element is the ratio of the potential difference across the element to the current flowing through the element. For a pure resistor this is purely real, i.e. there is no imaginary part.

The complex impedance of a pure resistor is:   equation



The physical potential and current are the imaginary parts of the complex values:

equation

Because there is no initial phase in either quantity, the emf is in phase with the current, as shown in the graph.

equation

The Power dissipated in a Resistor

The instantaneous power is:

equation
equation

By inspection, the average of sinθ is zero and the average of sin2θ is a half.

Taking the average over one period:

equation


Effective or r.m.s. values

From above

equation

Here I (no subscript), is an effective D.C. current that gives the same heating effect as the A.C. current. Since

equation

I is the square root of the mean of the square of the current, or r.m.s. current.



The r.m.s. value calculation above does not give a universal result, as it will depend on the waveshape supplied by the source. For example, with the sinusoidal potential difference and resistor above:

equation


Since most A.C. meters measure the r.m.s. values of current and potential, unless stated otherwise, all values shown on circuit diagrams will be r.m.s. values. For example, the 240 V domestic supply has a nominal maximum (or peak) value of 339 V ( = 240×√2 ). This goes up and down with sudden variations in demand, e.g. when most people leave their employment and go home.



Example
The 240 V, 50 Hz mains electricity supply is connected in series with a 1 kW heater. Find the rms and instantaneous values of
(a) the p.d. across the heater, and
(b) the current in the circuit, and
(c) the power dissipated in the heater.



In the following, upper case letters denote r.m.s. or steady D.C. values and lower case letters denote instantaneous values.

equation


A Pure Capacitor in an AC circuit

equation

The complex applied emf is Emexp[jωt].

The complex potential difference across the capacitor is equal to the complex applied emf.

equation


The instantaneous current is:

equation

Multiplying by j is a rotation of 900, so the current is ahead of the potential by 900 of phase.

The complex impedance( Z ) of the capacitor is:

equation


The complex impedance ( Z ) of the capacitor is purely imaginary, and because it has no real part to dissipate energy as heat, it is called a reactance ( X ).



The physical potential and current are:

equation


In brief:
• The current is 900 ahead of the applied potential difference, as shown in the graphs.

equation

• The current is limited by the capacitance and the angular frequency.

equation

Where XC is called the capacitative reactance. It is not a resistance but it has units of Ohm because it is a ratio of potential to current and limits the current.



The Power in a capacitor

The instantaneous power in a capacitor is given by:

equation

The instantaneous power changes at twice the frequency that is applied.


The average power will be zero because the average of a sine function is zero



Example
The 240 V, 50 Hz mains electricity supply is connected in series with a 5μF capacitor. Find the rms and instantaneous values of
(a) the current in the circuit, and
(b) the potential difference across the capacitor.



equation

equation

equation


A Pure Inductor in an AC circuit

equation

The complex applied emf is Emexp[jωt].

The complex potential difference across the inductor is equal to the complex applied emf.

equation


The integration constant is a possible steady D.C. current. It is set equal to zero here because there is no steady D.C. potential. Dividing by j is equivalent to multiplying by -j ( since 1/j×j/j = -j ). Multiplying by -j is a clockwise rotation of 900 in phase, so the current will lag the emf by 900.

The current in circuit is given by:

equation


The impedance of the inductor is given by:

equation

The complex impedance (Z) of the inductor is purely imaginary, and is also called a reactance (X).

The physical potential and current are:

equation


In brief:
• The current is 900 behind the applied potential difference.

equation
• The current is limited by the inductance and the angular frequency.

equation, where equation is called the inductive reactance. As before, it is not a resistance because inductors do not dissipate heat. Neverthless it has units of Ohm because it is a ratio of potential to current and limits current.



Example
The 240 V, 50 Hz mains electricity supply is connected in series with a 200mH inductor. Find the rms and instantaneous values of
(a) the current in the circuit, and
(b) the p.d. across the inductor.



equation

equation

equation


Summarising:

The average or D.C value: equation

The potential difference across a resistor is in phase with the current through it.

The average power dissipated in a resistor: equation

The effective or r.m.s. value of a sinusoidal waveform: equation

The complex impedance of a pure capacitor: ZC = XC = -j(1/ωC). The potential difference across a capacitor is 900 behind the phase of the current through it. The average power in a capacitor is zero

The complex impedance of a pure inductor: ZL = XL = j·ωC. The potential difference across an inductor is 900 ahead of the phase of the current through it. The average power in an inductor is zero

Current reference mnemonic:

equation

"level in the Room, down to the Cellar, up to the Loft".
The Resistor p.d. is level (=in phase) with the current,
The Capacitor p.d. is lower (=900 behind) in phase,
The Inductor p.d. is higher (=900 advanced) in phase.



Acknowledgement: These notes are based in part on "Alternating Current Circuit Theory" by G.J.Russell and K.Mann NSWUP 1969.


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